Identification of antibiotic targets and critical points in metabolic networks based on pathway analysis

ABSTRACT

A computational approach to identifying potential antimicrobial drug targets based on the structural capabilities of the microbe&#39;s metabolic network, which may be reconstructed from genomic and biochemical information. Starting with a cellular metabolic network (i) a stoichiometric matrix is generated to describe the connectivity of the reaction in the network, where (ii) constraints can be placed on various fluxes to allow for defined inputs and outputs to the network. For the defined network the unique set of extreme pathways can together be used to describe the complete range of metabolic capabilities of the network. From these pathways, sets of reactions whose elimination from the network removes certain production capabilities from the network can be mathematically determined by process of convex analysis.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention generally concerns the identification of(i) pathways and (ii) critical points in, and (iii) the generation ofmathematical models of, existing and proposed cellular metabolicnetworks comprised of biochemical reactions or mechanisms with geneticor non-genetic associations.

[0003] More specifically, the present invention relates to computationalmethods and systems for the analysis and modeling of cellular metabolicnetworks so that, inter alia, potential targets in support of thedirected development of therapeutic agents and engineered microbialstrains may be identified.

[0004] 2. Description of the Prior Art

[0005] 2.1 General Background

[0006] Within a cell of any organism there are complicated networks ofinteracting proteins and enzymes that perform certain chemicalconversions and transformations. These conversions andtransformations—life processes—ultimately lead to the production of the(i) necessary building blocks (biomass constituents such as amino acids,nucleotides, phospholipids, etc.) and (ii) energy requirements of thecell. Environmental substances are processed to meet the demands of aliving cell through this the cell's network of biochemical reactions.

[0007] These biochemical reaction networks primarily involve the use ofenzymes derived from particular genes whose chromosomal location andfunction have been characterized, as well as enzymes inferred to bepresent based on similarity of their genomic sequence to the genomicsequences of enzyme-coding genes in other organisms. There is presently,circa 2000, much focus on attempting to model and to ‘reconstruct’ thesenetworks of a living organism based, primarily, on the use of genomesequence information of the organism.

[0008] Meanwhile, the arsenal of reactions that a cell has at itsdisposal dictate the production capabilities and maximal performancecharacteristics of the cell. To change these capabilities the cell wouldhave to acquire new biochemical reactions through some evolutionarymechanism. In so doing the cell would of necessity increase the range offeasible routes by which it could meet certain cellular demands from aset of environmental supplies.

[0009] 2.2 The Utility of Mathematics to Analyze Biochemical ReactionNetworks

[0010] The capabilities of cellular biochemical reaction networks (i) toproduce necessary building blocks and energy requirements, and (ii) toevolve the reaction pathways by which cellular production(s) is (are)realized, can be comprehensively examined using rigorous mathematics.Mathematical examination yields results which are biochemicallymeaningful, serving to predict the performance of the biochemicalreaction network.

[0011] There exists one particular type of mathematical analysis ofcellular biochemical reaction networks called “convex analysis”. needdefinition. Some of the principles of convex analysis were previouslyused by Schuster to find “elementary nodes”, or reactions within thebiochemical reaction networks. See Schuster, S., T. Dandekar and D. A.Fell, 1999, Detection of elementary flux modes in biochemical networks:a promising tool for pathway analysis and metabolic engineering, TrendsBiotechnology 17(2): 53-60. See also Schuster, S. and C. Hilgetag, 1994,On elementary flux modes in biochemical reaction systems at steadystate. J. Biological Systems 2(2): 165-182. Finally see Schuster, S., C.Hilgetag, J. H. Woods and D. A. Fell, 1996, Elementary modes offunctioning in biochemical networks.

[0012] Convex analysis was also previously used by Bruce Clarke to find“extreme currents”, or reaction pathways through the biochemicalreaction networks by which pathways the biochemical reaction networkssucceed in processing environmental substances into the building blocksand energy requirements of the cell. See Clarke, B. L., 1980, Stabilityof Complex Reaction Networks. Advances in Chemical Physics 43: 1-215.See also Clarke, B. L., 1981, Complete set of steady states for thegeneral stoichiometric dynamical system. J. Chem. Phys. 75(10):4970-4979.

[0013] The mathematics associated with convex analysis may be used todetermine the minimal set of biochemical pathways by which someparticular capability of the biochemical reaction network is realized.These pathways satisfy both (i) mass balance constraints (associatedwith stoichiometry) and (ii) directional constraints placed on reactions(associated with thermodynamics).

[0014] These pathways are termed “extreme pathways”, and canbeneficially be used to examine the functional capabilities of abiochemical reaction network. Importantly, from knowledge of theseextreme pathways it is possible to determine all of the possiblecombinations of reactions that need to be eliminated from the network toremove some particular capability(ies) of the network. From the lists ofreactions it is a simple step to determine the enzymes and genesresponsible for these reactions.

[0015] Consider now that the elimination of these genes should thenrender the biochemical reaction network incapable of reaching someparticular outcome, some particular demand(s) of the cell!

[0016] Conversely, once it is understood what a biochemical reactionnetwork is doing, and how it is doing it, then it may become possible to“re-engineer” the network, and the organism, to steer more of its outputinto desired channels (i.e., to make more of a desired reactionproduct).

[0017] Mathematical tools that permit recognition of pathways withinbiochemical reaction networks, and of the genes involved with thereactions within these pathways, have still further implications for thedevelopment of antibiotics to combat microbial infections. The toolspermit recognition of how a deleterious process and pathway of thebiochemical reaction network might be stopped, or at least disrupted.

[0018] Alternatively, the same mathematical computational tools can beused to improve the design and engineering of organisms for industrialapplication such as the production of bio-commodities. The tools permitrecognition of how an beneficial process and pathway of the biochemicalreaction network might be augmented or accentuated.

[0019] 2.3 Specific Prior Art Mathematical Analysis of BiochemicalReaction Networks

[0020] Convex analysis has been previously used to study biochemicalsystems and to generate related sets of pathways called elementary modesand extreme currents. For a comprehensive review see the paper byinventor of the present invention Schilling and his colleagues, seeSchilling, C. H., S. Schuster, B. O. Palsson and R. Heinrich, 1999,Metabolic pathway analysis: basic concepts and scientific applicationsin the post-genomic era. Biotechnology Progress 15(3): 296-303.

[0021] Elementary modes have been used to metabolically engineerbacteria for producing aromatic amino acid precursors at yields near themaximum theoretical yield. See Liao, J. C., S-Y Hou and Y-P Chao, 1996,Pathway Analysis, Engineering, and Physiological Considerations forRedirecting Central Metabolism.

[0022] Liao, et al., report research where all of the elementary modesfor a reduced reaction network in Escherichia coli were calculated andstudied to determine the optimal flux distributions through a centralmetabolism that redirected carbon flow to the pathways from aromaticamino acid production. Reactions that did not appear in the optimalpathways were considered indispensable, while those that did appear inthe optimal pathways were candidates for over-expression.

[0023] A similar analysis can be performed with the extreme pathwaysrather than elementary modes. For the precise difference between thesetwo approaches see Schilling, C. H., D. Letscher and B. O. Palsson,2000, Theory for the systemic definition of metabolic pathways and theiruse in interpreting metabolic function from a pathway-orientedperspective. Journal of Theoretical Biology 203(3): 229-248.

[0024] Still other papers and publications discuss the complex analysisof biochemical reaction networks. See (i) Clarke, B. L. 1980, Stabilityof Complex Reaction Networks, Advances in Chemical Physics 43: 1-215;(ii) Clarke, B. L., 1981, Complete set of steady states for the generalstoichiometric dynamical system, J. Chem. Phys. 75(10): 4970-4979; (iii)Edwards, J. S., R. Ramakrishna, C. H. Schilling and B. O. Palsson, 1999,Metabolic flux balance analysis, (iv) Metabolic Engineering, S. Y. Leeand E. T. Papoutsakis, New York, Marcel Decker, Inc.: 13-58; (v) Liao,J. C., S-Y Hou and Y-P Chao, 1996, Pathway Analysis, Engineering, andPhysiological Considerations for Redirecting Central Metabolism,Biotechnology and Bioengineering 52: 129-140; (vi) Schilling, C. H., S.Schuster, B. O. Palsson and R. Heinrich, 1999, Metabolic pathwayanalysis: basic concepts and scientific applications in the post-genomicera. Biotechnology Progress 15(3): 296-303; (vii) Schuster, S., T.Dandekar and D. A. Fell, 1999, Detection of elementary flux modes inbiochemical networks: a promising tool for pathway analysis andmetabolic engineering, Trends Biotechnology 17(2): 53-60; (viii)Schuster, S. and C. Hilgetag, 1994, On elementary flux modes inbiochemical reaction systems at steady state, J. Biological Systems2(2): 165-182; (ix) Schuster, S., C. Hilgetag, J. H. Woods and D. A.Fell, 1996, Elementary modes of functioning in biochemical networks; (x)Computation in Cellular and Molecular Biological Systems, R.Cuthbertson, M. Holcombe and R. Paton, London, World Scientific:151-165; and (xi) Varma, A. and B. O. Palsson, 1994. Metabolic FluxBalancing: Basic concepts, Scientific and Practical Use. Bio/Technology12: 994-998.

SUMMARY OF THE INVENTION

[0025] The present invention contemplates improvements to the existingmathematical method of convex analysis for purposes of analyzing theproduction of one or more selected metabolites of a biochemical reactionnetwork so producing metabolites. In simple terms, the improvedmathematical method of the present invention explains why a biochemicalreaction network—also known as a cellular metabolic network—does what itdoes. The utility of so knowing is, of course, that precisionunderstanding of the processes of nature best supports the effective,useful and safe manipulation of these processes. For example, abiochemical process inducing disease may sometimes usefully be stopped;a biochemical process producing a valuable biochemical as an metabolicoutput may sometimes usefully be enhanced.

[0026] The present invention contemplates still more than gainingknowledge of a biochemical reaction network: the present inventionfurther contemplates an improved application of the mathematical processof convex analysis particularly for the purpose of identifying criticalpoints in cellular metabolic networks engaging in biochemical reactions.The improved analysis proceeds so that, by identification of somecritical point, all members of the particular set of complex pathways ofbiochemical reactions leading to this point may be better understood.The better understanding this set of complex pathways again permits thebetter understanding of which reactions, and associated reactionpathways, can either be (i) disrupted so as to defeat that the cellularmetabolic network should attain this critical point, or (ii) enhanced,so that the cellular metabolic network will produce more of someselected biochemical(s) at the critical point, and/or produce thisbiochemical (these biochemicals) faster. In simple possible terms, theimproved mathematical method of the present invention not only explainshow a biochemical reaction network, or cellular metabolic network,functions (as in the preceding paragraph), but permits analysis of whyand how the network might be (i) precluded or obstructed from working(as in the avoidance of disease), or, alternatively (ii) made to workbetter and/or faster (as in the production of a useful biochemicalproduct).

[0027] The present invention contemplates yet still more: an improvedmethod of deriving a mathematical system of linear equations and linearinequalities representing—i.e., serving as a mathematical model of—abiochemical reaction network (also known as a metabolic network) inother that the derived system may be rigorously analyzed. Improvementsto the existing mathematical process of convex analysis are againinvolved. In simplest possible terms, the present invention deliversmore than just a “road map” to biochemical life processes (as in thesecond preceding paragraph), or even a “road map” with all major “waypoints” prominently accurately identified (as in the immediatelypreceding paragraph), but actually delivers a mathematical model of thebiochemical reactions of a metabolic network which mathematical modelquantitatively captures the production capabilities of the network.

[0028] Clearly such knowledge of what is produced, by what processes,when, and to what amount(s) starts to subject life processes—as areconducted by cellular metabolic networks engaging in biochemicalreactions—to the techniques of (bio)chemical engineering. One the(bio)chemical “plant” and its processes are understood, then a(bio)chemical engineer should, and is, able to approach modification ofthe “plant” and/or its (life) processes in order to obtain different, orbetter, results.

[0029] 1. General Approach of the Present Invention

[0030] The present invention provides a general framework and system forthe identification of all the minimal sets of reactions that, whenremoved from a biochemical reaction network, will render the networkunable to reach its particular production objectives.

[0031] As precursor steps to the present invention, biochemical reactionnetworks are conventionally (i) constructed from genomic and biochemicaldata and (ii) described by a stoichiometric matrix. Together, theconstraints on the directions in which reactions can proceed and thestoichiometric matrix correspond to a mathematical system of linearequations and linear inequalities, which system can be studied usingconvex analysis.

[0032] In accordance with the present invention, sets of extremepathways are defined that are used to represent all of the possiblesteady states which the network can achieve. By removing a singlereaction in the network all of the pathways that utilize this reactionare also removed. To remove the ability of the network to reach aparticular objective, all of the extreme pathways that reach thisobjective must be removed.

[0033] In one embodiment of the invention, a set of reactions and theirassociated genes are identified that will eliminate the ability of thetarget organism to generate essential biomolecules necessary for itsgrowth.

[0034] Rather than removing one reaction at a time—or many reactions ata time—in order to determine the effects of the removal(s) on theperformance of the reaction network, it is ideal to be able to calculateall of the sets of reactions that will eliminate the capability of thenetwork to achieve a particular production objective. In an additionalembodiment of the invention, an algorithm is presented by which it ispossible to generate the entire group of sets of reactions that can beused to eliminate functional properties and network capabilities ofinterest in a target organism. Collectively this group of reaction setswill be called “minimal deletion sets”.

[0035] These deletion sets are unique to the present invention, andquite valuable. They may be, for example, used to identify reactionswhich are critical to the performance of the network under a particularcondition, but whose function is essentially redundant under otherconditions. In addition to finding global deletion sets that arecritical under all conditions it is then possible to find environmentalor condition-specific deletion sets.

[0036] In summary, beginning with an advanced, but conventional,stoichiometric matrix and a set of general reaction constraints, thepresent invention shows how to generate (i) the set of extreme pathwaysalong with (ii) the process for identifying a minimal deletion sets fromthese pathways. An (iii) algorithm is then shown by application of whichalgorithm all the minimal deletion sets in the network may be may beidentified.

[0037] These mathematical processes of the present invention, asdiscussed in greater detail hereinafter, are normally embodied in acomputer software program. The program can be used to calculate theextreme pathways and minimal deletion sets of a biochemical reactionnetwork of virtually any complexity. These valuable minimal deletionsets can then be used, for example, in the subsequent development ofdrugs to combat infectious disease. They may alternatively be used forthe rational design and engineering of organisms for the production ofbiomolecules of interest (often referred to as metabolic engineering).

[0038] 2. A Method of Analyzing the Production of One or More SelectedMetabolites of a Biochemical Reaction Network Producing Metabolites

[0039] Accordingly, in one of its aspects the present invention isembodied in an improvement to an existing method of analyzing theproduction of one or more selected metabolites of a biochemical reactionnetwork that produces metabolites.

[0040] The existing method has as inputs (i) reactions of thebiochemical reaction network constructed from genomic and biochemicaldata, (ii) exchange fluxes on such of the produced metabolites as are ofinterest as inputs and outputs to the network, (iii) a stoichiometricmatrix, developed from the reactions in consideration of the exchangefluxes, defining participation of each network metabolite in eachreaction and exchange flux of the network, and (iv) a system of linearequations and inequalities mathematically defining the network. Fromthese inputs the existing method serves to identify deletion sets ofreactions that, when removed from the network, eliminate the capabilityof the network to produce a selected metabolite.

[0041] In particular, the improvement to this existing method inaccordance with the present invention uses the linear equations andinequalities of the network to mathematically calculate a convexsolution space called a “flux cone”. The calculating produces“generating vectors” of this flux cone, which generating vectors arecalled “extreme pathways”. Using these generating vectors (calledextreme pathways), the mathematical process of the present inventioncontinues to determine sets of reactions that, when deleted, diminish oreliminate capability of the network to produce an output metabolite ofinterest.

[0042] These mathematically determined sets correspond to criticalreactions of the network which, when stopped, affect the capability ofthe network to produce the output metabolite of interest.

[0043] The method of the present invention may, after the determining ofsets of reactions, optionally continue with selecting from thedetermined sets of reactions those sets that totally eliminate thecapability of the network to produce the output metabolite of interest.In this case the selected sets are called “deletion sets” becausedeletion of the reactions represented by the pathways of these setssuffices to totally eliminate the production of the output metabolite ofinterest by the network.

[0044] The calculating of the generating vectors of the flux conepreferably ensues by specific mathematical manipulations within the moregeneral mathematical process of convex analysis. More particularly, thespecific preferred convex analysis of the present invention consists ofcalculating any of (i) a conical basis, (ii) a convex basis, (iii) alinear basis, or (iv) a combination of any of conical and convex andlinear bases.

[0045] Preferably in the method of the present invention at least someof the constructed reactions will have an associated constraint upon thedirection in which the reaction can proceed.

[0046] The method of the present invention may be used, for example, toproduce an output of interest which consists of one or more functionalproperties of interest in the analyzed biochemical production network.In this case the reaction sets show how these one or more functionalproperties of interest can be diminished or eliminated. If the output ofinterest consists of, for example, but one single functional property ofinterest in the analyzed biochemical production network then thereaction sets show how this functional property of interest can bediminished or eliminated.

[0047] For example, that biochemical reaction network which is analyzedby the mathematical method of the present invention can represent adisease-producing, pathogenic, organism. The metabolite of interest willbe one that is necessary for survival of the pathogenic organism. Inthis situation the method of the present invention using the reactionset can be directed to targeting development of a drug that, byobstructing those reactions of the pathogenic organism that produce themetabolite necessary for survival of the organism, serves to eliminatethe pathogenic organism.

[0048] As another example, the biochemical reaction network that isanalyzed can again represent a disease-producing, pathogenic, organism.However, this time the metabolite of interest can be the actualdisease-producing, deleterious, substance that is produced by thepathogenic organism. In this situation the method of the presentinvention using the reaction set can be directed to targeting thedevelopment of a drug that, by obstructing those reactions of thepathogenic organism that produce the metabolite that induces disease,serves to eliminate the deleterious, disease-causing, function of thepathogenic organism.

[0049] As an oppositely-directed example, when the reaction networkanalyzed is an organism producing both desired bio-molecules of valueand un-desired bio-molecules of no value, and when the metabolite ofinterest produced by the organism is defined to be the un-desired andvalueless bio-molecules, then the method of the present invention usingthe reaction set can be directed to metabolically re-engineering theorganism to fail of those reactions that produce the particularmetabolite that is un-desired and valueless. Thus production ofun-desired valueless bio-molecules can be eliminated while continuedproduction of desired valued bio-molecules is permitted.

[0050] Similarly to this example, when the reaction network analyzed isan organism producing desired bio-molecules of value by each of two ormore—multiple—metabolic routes, and when the metabolite of interest isdefined to be the valued molecule as is produced by oneonly—preferred—route of the multiple routes by which the organism iscapable of producing this molecule, then the method of the presentinvention using the reaction set can be directed to metabolicallyre-engineering the organism to fail of those reactions that produce themetabolite of interest via inefficient route(s), therein by eliminatingproduction of metabolite via this route (these routes) nonetheless thatthe metabolite is of value. Production of the desired metabolite by oneor more alternative one(s) of the multiple metabolic routes is leftintact, and may even be accentuated.

[0051] These combinations show the reaction set produced by themathematical method of the present invention to be a valuable tool. Insimple terms, the reaction set shows how to preclude, or to obstruct, orto accentuate individual biochemical pathways within the organism aslead to the production of particular metabolites. It is hard to ask formore than this: a complete quantitative, mathematical, model as to thebiochemical reactions of the cell.

[0052] 3. A Method of Identifying Critical Points in Cellular MetabolicNetworks Engaging in Biochemical Reactions

[0053] In another of its aspects the present invention is embodied in amethod of (i) applying in an improved way the existing mathematicalprocess of convex analysis so that a convex hull is defined and spannedby unique generating, or edge, vectors, this hull being analyzed toderive a particular solution that is, mathematically, a particular pointdescribed by a flux vector lying within the interior of the convex hull.All this effort to get to this particular solution is for the purpose of(ii) identifying critical points in cellular metabolic networks engagingin biochemical reactions so that, by identification of some criticalpoint, the particular set of complex pathways of biochemical reactionsleading to this point may be better understood. All this effort to getto the understanding of all the biochemical reactions leading to aparticular point is so that, (iii) by better understanding this set ofcomplex pathways, it may further be better understood which reactions,and associated pathways, can be selectively disrupted so as to defeatthat the cellular metabolic network should attain this critical point.

[0054] The method of the present invention thus consists of using thisconvex hull—a mathematical construction—to represent the capabilities ofa metabolic genotype. By this usage the unique generating, edge, vectorsthat define and that span the convex hull represent systemicallyindependent extreme pathways of the metabolic, life, processes of themetabolic genotype.

[0055] Likewise in this usage every point in the hull is somenon-negative combination of the unique generating, edge, vectorscorresponding to the fact that every metabolic, life, process of themetabolic genotype is some combination of the extreme pathways of thesemetabolic processes.

[0056] Next in the method of the present invention, the convex hull ismathematically solved, again by a specific application of the moregeneral process of convex analysis, so as to derive a particularsolution that represents a metabolic phenotype. This particular solutionis, mathematically, a particular point described by a flux vector lyingwithin the interior of the convex hull.

[0057] The mathematical solving is repeated until a complete set ofparticular solutions, corresponding to a set of flux vectors each lyingwithin the convex hull, is derived. This set of solutions corresponds toall the pathways by which a particular metabolic phenotype is realized.

[0058] Thus the derivation of all pathways by which the particularmetabolic phenotype is realized is tantamount to recognition of all thebiochemical reactions that, as part of any pathways, lead to theparticular metabolic phenotype. Moreover, recognition of all biochemicalreactions variously leading to the particular metabolic phenotypepermits better understanding of what biochemical reactions of themetabolic genotype can be in particular disrupted so as to cause thatthe metabolic genotype should be unable to realize the particularsolution.

[0059] For example, this the method of the present invention can beemployed on the genotype of a pathogenic, disease-causing, organism. Inthis case the method of the present invention preferably continues withthe development of drugs that, by obstructing those biochemicalreactions of the genotype of the pathogenic organism that lead to aparticular, disease-inducing, solution of the genotype, serve toeliminate the deleterious, disease-causing, phenotype of the pathogenicorganism.

[0060] For example, this the method of the present invention can beemployed on the genotype of an organism producing both (i) desiredbio-molecules of value and (ii) undesired bio-molecules of no value. Inthis case the method of the present invention preferably continues withmetabolically re-engineering the organism so as to obstruct thosebiochemical reactions of the genotype of the pathogenic organism thatlead to that particular solution where the phenotype produces theundesired valueless bio-molecules, eliminating production of theseundesired valueless bio-molecules while permitting continued productionof desired valued bio-molecules.

[0061] 4. A Method of Analyzing a Metabolic Network

[0062] In still another of its aspects the present invention is embodiedin a method of analyzing a metabolic network.

[0063] The method consists of first identifying all biochemicalreactions occurring in the metabolic network, including any directionsthereof. Then all exchange fluxes are specified, including anyassociated directional restraints attendant upon metabolites of theidentified biochemical reactions.

[0064] A stoichiometric matrix where each column in the matrixcorresponds to a reaction, or flux, and where each row corresponds to adifferent metabolite involved in the metabolic network is next created.This created stoichiometric matrix represents, in all its columns androws, the collective biochemical reactions, being a form of chemicalconversion, and the collective cellular transport processes of themetabolic network, which cellular transport processes are how themetabolites enter and leave the metabolic network.

[0065] All directional constraints on the exchange fluxes are nextcombined with the created stoichiometric matrix to define the metabolicnetwork as a system of linear equations and linear inequalities.

[0066] Finally the system of linear equations and linear inequalitiesthat jointly define the metabolic network are analyzed by themathematical process of convex analysis.

[0067] The metabolic network defined as a system of linear equations andlinear inequalities may particularly obeys the three equations

S·v=0  (Equation 1)

[0068] where S refers to the stoichiometric matrix of the system and vis the flux vector;

v_(i)≧0, ∀i  (Equation 2)

[0069] where v_(i) corresponds to the flux value of the i^(th) reaction;and

α_(i)≦b_(i)≦β_(i)  (Equation 3)

[0070] where α_(i) and β_(i) are ether zero of negative and positiveinfinity, respectively, based on the direction of exchange flux, andb_(i) corresponds to the i^(th) exchange flux.

[0071] In this case the analyzing consists of solving the equations 1-3in convex space as a convex polyhedral cone in n-dimensional spaceemanating from the origin of the space.

[0072] Every point on the convex polyhedral cone may in particular berepresented by $\begin{matrix}{{C = {{v:v} = {\sum\limits_{l - 1}^{k}{\omega_{i}p_{i}}}}},{\omega_{i} \geq {0\forall_{i}}}} & \left( {{Equation}\quad 4} \right)\end{matrix}$

[0073] In this case the analyzing consists of calculating the conicalhull of the flux cone as representing the extreme pathways in themetabolic network.

[0074] After this calculating (which is a part of the analyzing), themethod may further continue by determining from the calculated pathwayscritical biochemical reactions, or sets of biochemical reactions, thatare required for the metabolic network to attain a particular objectiveor group of objectives—as is (are) represented by one or more particularpoints on the flux cone.

[0075] These and other aspects and attributes of the present inventionwill become increasingly clear upon reference to the following drawingsand accompanying specification.

BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Referring particularly to the drawings for the purpose ofillustration only and not to limit the scope of the invention in anyway, these illustrations follow:

[0077]FIG. 1 is a flow diagram illustrating one procedure of the presentinvention for determining one set, or the entire collection, of minimaldeletion sets that eliminate particular production capabilities ofinterest in a metabolic network.

[0078]FIG. 2 is a geometric representation of the flux cone of themathematical method of the present invention shown in three-dimensionswhere the entire unbounded flux cone is spanned by the generatingvectors representing the capabilities of a metabolic genotype.

[0079]FIG. 3, consisting of FIGS. 3a-3 c, are respectively a graph of anexemplary metabolic reactions scheme, a legend to the mathematicalrepresentation of the scheme, and table of the extreme pathways ascollectively define a metabolic genotype and phenotype in the context ofconvex analysis, where, more particularly,

[0080]FIG. 3(a) shows a hypothetical reaction network comprised of aseries of internal and exchange fluxes, functioning to generateappropriate ratios of metabolites C, D, and E for incorporation intobiomass represented as the GRO metabolite;

[0081]FIG. 3(b) shows a mathematical translation of the reaction networkinto the steady-state mass balances and constraints placed on all fluxesthat define the solution domain; and

[0082]FIG. 3(c) is a list of the ten (10) extreme pathways calculatedfor the network shown in vector format indicating the relative fluxactivity of each of the fluxes in the pathways.

[0083]FIG. 4 is a table 1 showing the set of extreme pathways *p1, . . ., p10) for the reaction scheme shown in FIG. 3.

[0084]FIG. 5 is a graph showing normalized values for the objective flux(b_(z)) for all single and double deletion combinations of the networkdescribed in FIG. 3.

DESCRIPTION OF THE PREFERRED EMBODIMENT

[0085] Although specific embodiments of the invention will now bedescribed with reference to the drawings, it should be understood thatsuch embodiments are by way of example only and are merely illustrativeof but a small number of the many possible specific embodiments to whichthe principles of the invention may be applied. Various changes andmodifications obvious to one skilled in the art to which the inventionpertains are deemed to be within the spirit, scope and contemplation ofthe invention as further defined in the appended claims.

[0086] The present invention relates to systems and methods for (i)identifying deletion sets in a target organism's metabolic network (ii)based on a pathway analysis (iii) mathematically derived from (iv) alist of reactions associated with convexity, which reactions are apartial or a complete representation of reactions in a cell.

[0087] A stoichiometric matrix is used to describe the connectivityamongst all the metabolites in a metabolic network of any complexityincluding that for an entire organism. Additional constraints on thedirection in which reactions and fluxes can proceed in the network serveto complete the mathematical description of the reaction network. Fromthis description existing algorithms can be used to calculate the set ofextreme pathways that form the conical hull of the solution space,corresponding to all feasible steady state flux distributions. Thedetails of performing the construction of a stoichiometric matrix andthe formulation of the constraints imposed on a biochemical reactionnetwork have been previously disclosed along with an algorithm forcalculating the set of extreme pathways (Schilling, 2000). From thesepathways it is possible to identify sets of reactions that upon removalfrom the network eliminate the ability of the network to achieveselected objective of interest. Moreover, it is possible to calculateall of these deletion sets using one algorithm that operates on the setof extreme pathways. These minimal deletion sets may be useful for theidentification and development of potential protein and genetic targetsfor anti-microbial drugs and the engineering of microbial strains forbioproduction purposes.

[0088] It should be noted that the systems and methods described hereincan be implemented on any conventional host computer system, such asthose based on Intel® microprocessors and running Microsoft Windowsoperating systems. Other systems, such as those using the UNIX or LINUXoperating system and based on IBM®, DEC® or Motorola® microprocessorsare also contemplated. The systems and methods described herein can alsobe implemented to run on client-server systems and wide-area networks,such as the Internet.

[0089] Software to implement the system can be written in any well-knowncomputer language, such as Java, C, C++, Visual Basic, FORTRAN or COBOLand compiled using any well-known compatible compiler.

[0090] The software of the invention normally runs from instructionsstored in a memory on the host computer system. Such a memory can be ahard disk, Random Access Memory, Read Only Memory and Flash Memory.Other types of memories are also contemplated to function within thescope of the invention.

[0091] The process 10 for determining all minimal deletion sets for anymetabolic network of interest is shown in FIG. 1. Beginning at a startstate 12, the process 10 then moves to a state 14 to gather all of theknown biochemical information pertaining to the reactions in thenetwork. This generates a list of all the chemical reactions occurringin the network. This provides the stoichiometry of each chemicalreaction proposed to occur in the network along with informationregarding the irreversible and reversible nature of the reactions. Thestoichiometry of each reaction provides the molecular ratios in whichreactants are converted into products. These reactions include alltransport reactions, enzymatic reactions, and diffusion processes. Foran entire organism these reactions are catalyzed by gene products whosepotential presence in the organism can be inferred from the annotatedgenome sequence of an organism along with additional biochemicalinformation. Thus the reactions determined in state 14 represent all ofthe physical-chemical conversions that can reasonably occur in thesystem.

[0092] A theoretical system boundary can be drawn around all of thephysically occurring reactions and associated substrates and productsdefined in state 14. These reactions contained within the systemboundary will be collectively referred to as the internal fluxes of thesystem. The entire group of substrates and products are collectivelyreferred to as the metabolites of the system. The process 10 then movesto a state 16 wherein all of the exchange fluxes are specified in thesystem. These exchange fluxes will constitute the presence of potentialsources and/or sinks on individual metabolites. Therefore if aparticular metabolite is to be allowed to enter the system or exit thesystem an exchange flux is created to allow for the passage of themetabolite across the theoretical system boundary. Conceptually thesefluxes can be thought of as the input and outputs of the system and canbe defined by the researcher to simulate environmental conditions ofinterest.

[0093] The information obtained in state 14 and 16 regarding thereactions and metabolites participating in the system must be translateinto a mathematical format to facilitate the remainder of the process10. While reversible reactions may be represented as one reaction, forease of calculation all reactions that are reversible will be decomposedinto a forward reaction and a backward reaction that are constrained toonly proceed in one positive direction. Therefore all internal fluxeswill be constrained to take on only positive values representing theiractivity levels. Exchange fluxes that serve as both input and outputsfor a particular metabolite may also be decomposed in a similar fashionor remain bi-directional, thus capable of taking on any real value(positive and negative). All of the information regarding the internalfluxes, exchange fluxes, and their substrates and products can berepresented in a matrix format typically referred to as a stoichiometricmatrix. Each column in the matrix corresponds to a given reaction orflux, and each row corresponds to the different metabolites involved inthe system. Thus, a given position in the matrix describes thestoichiometric participation of a metabolite (listed in the given row)in a particular flux of interest (listed in the given column). Togetherall of the columns of the stoichiometric matrix represent all of thechemical conversions and cellular transport processes that aredetermined to be present in the network. This includes all internalfluxes operating within the system and exchange fluxes operating on thesystem as inputs and outputs. Thus, the process moves to a state 18 inorder to formulate all of the fluxes into a stoichiometric matrix.

[0094] Next, the process 10 moves to a state 20 wherein informationregarding the directional constraints placed on the fluxes is combinedwith the stoichiometric matrix to fully define the metabolic network asa system of linear equations and linear inequalities to be analyzedusing principles of convex analysis. In studying metabolic networks theprinciple of conservation of mass is applied in the generation oftransient mass balances around each metabolite in the system. Each massbalance constitutes a differential equation describing the change inconcentration of the metabolite as function of the activities of thefluxes that serve to generate and dissipate the metabolite. Structuralaspects of a metabolic network are time invariant allowing thefunctioning of the network to be placed into a steady state. Eliminatingall the time derivatives obtained from dynamic mass balances aroundevery metabolite in the metabolic system, yields the system of linearequations represented in matrix notation,

S·v=0  (Equation 1)

[0095] where S refers to the stoichiometric matrix of the system, v isthe flux vector. This equation simply states that over long times, theformation fluxes of a metabolite must be balanced by the degradationfluxes. Otherwise, significant amounts of the metabolite will accumulateinside the metabolic network. Applying equation 1 to our system we let Snow represent the stoichiometric matrix constructed in state 18.

[0096] To completely describe the metabolic system it is necessary toinclude the constraints on the possible directions of the internal andexchange fluxes. The constraints on the internal fluxes is ratherstraightforward as all fluxes must be non-negative yielding:

v_(i)≧0, ∀i  (Equation 2)

[0097] where v_(i) corresponds to the flux value of the i^(th) reaction.The constraints on the exchange fluxes depend on the status of thedetermined source or sink on the associated metabolite, or similarly onthe input and output status of the metabolite. This can be expressed inEquation 3 below where α_(i) and β_(i) are either zero or negative andpositive infinity, respectively, based on the direction of the exchangeflux and b_(i) corresponds to the i^(th) exchange flux.

α_(i)≦b_(i)≦β_(i)  (Equation 3)

[0098] Under the existence of a source(input) only α_(i) is set tonegative infinity and β_(i) is set to zero, whereas if only asink(output) exists on the metabolite α_(i) is set to zero and β_(i) isset to positive infinity. If both a source and sink are present then theexchange flux is bi-directional with α_(i) set to negative infinity andβ_(i) is set to positive infinity leaving the exchange fluxunconstrained.

[0099] Together equations 1-3 describe the metabolic system understeady-state conditions as a system of linear equalities and linearinequalities. The presence of linear inequalities limits the use oftraditional concepts of linear algebra, and necessitates the use ofconvex analysis, which is capable of treating systems of linearinequalities. The set of solutions to any system of linear inequalitiesincluded that described above is a convex space. This convex solutioncorresponds geometrically to a convex polyhedral cone in n-dimensionalspace (R^(n)) emanating from the origin for all metabolic systemsmodeled as described herein. We refer to this convex cone generally asthe flux space and more specifically as the steady-state flux cone (C).Within this flux cone lies all of the possible solutions and hence fluxdistributions under which the system can operate. Since every solutionor operating mode of the system is contained within the flux space, itlogically follows that the entire flux space represents the capabilitiesof the given metabolic network. Thus the flux space clearly defines whata network can and cannot do.

[0100] Convex cones are described by the extremal rays (or generatingvectors) that correspond to the edges of the cone, (being half-linesemanating form the origin). These extremal rays are said to generate thecone (forming the conical hull) and cannot be decomposed into anon-trivial convex combination of any other vectors residing in the fluxcone. Here in the context of metabolic systems the term extreme pathwaysis used to denote the extreme rays of a polyhedral cone as each raycorresponds to a particular pathway or active set of fluxes whichsatisfies the steady-state mass balance constraints and inequalities ofequations 1-3. Extreme pathways are denoted by the vector p_(i) and thetotal number of extreme pathways needed to generate the flux cone for asystem will be denoted by k. Every point within the cone (C) can bewritten as a non-negative convex linear combination of the extremepathways as shown below: $\begin{matrix}{{C = {{v:v} = {\sum\limits_{l - 1}^{k}{\omega_{i}p_{i}}}}},{\omega_{i} \geq {0\forall_{i}}}} & \left( {{Equation}\quad 4} \right)\end{matrix}$

[0101] Thus the set of extreme pathways is analogous to abasis/coordinate system that can be used to describe a position inspace. These pathways are said to conically span or generate the set ofall pathways as any pathway and/or distribution of fluxes can be writtenas a non-negative linear combination of the p_(i)'s. The pathway vectorw corresponds to the coordinate vector relative to the set of extremepathways. It provides the weight given to each pathway in a particularflux distribution (v).

[0102] The calculation of the unique set of extreme pathways for ametabolic network, described by equations 1-3, moves the process 10 to astate 22. These pathways can be calculated using any algorithm capableof generating the conical hull of a convex polyhedral cone. One suchalgorithm has been previously disclosed (Schilling, 2000).

[0103] The set of extreme pathways geometrically represent the edges ofthe flux cone as depicted in FIG. 2, where the flux cone is the regionof all admissible flux distributions that a metabolic network candisplay under the assumption of steady state. These pathways define thestructural capabilities of the network and can be used to interpretevery possible flux distribution or metabolic phenotype/functionalmodality of a metabolic network. Thus they define what a network can andcannot do; what building blocks can be produced; how efficient can thenetwork generate biomass or extract energy from substrates; what are theredundancies in the network.

[0104] Of particular interest is the ability to use these pathways toidentify critical reactions or sets of reactions that are required forthe network to reach a particular objective(s). These lethal or minimalsets of reactions and the genes coding for the gene products of thesereactions are herein referred to as minimal deletion sets. Furthermore,these pathways can be used to calculate every possible deletion set in anetwork for a given set of input and output conditions. Any combinationof genetic deletions that would be lethal to an organism must contain asubset of genes that form a minimal deletion set. Therefore if thecombined loss of gene A and B is lethal to an organism, where A and Balone are not essential genes, then the set is a minimal deletion set,making every superset of this also a deletion set (such as the loss ofgene A, B and C).

[0105] When considering the ability of a metabolic network to produce adefined set of metabolic precursors used to generate all of thecomponents of the biomass, these deletion sets correspond to targets orcombined targets for antimicrobial therapeutics. When considering thespecific production of a metabolite (i.e. amino acid) these setscorrespond to reactions that can be deleted to direct the flow ofmetabolic resources in the cell for the metabolic engineering of anorganism.

[0106] The identification of the complete group of minimal deletion setsfor a metabolic network moves process 10 to a state 24. The procedurefor determining a minimal deletion set follows a simple rationale. Whenan internal flux is eliminated from the network or identically if theflux value is forced to zero it is seen from Equation 4 that each of thepathways that utilize the particular flux are forced to be weighted tozero, otherwise non-zero values would exist in v corresponding to theeliminated reaction. Therefore all pathways that use this flux can beeliminated for further analysis of the situation. After removing theappropriate pathways, if there are no pathways capable of producing adesired output or objective then there exists no combination of pathwaysthat can generate the particular objective. To calculate the completegroup of minimal deletion sets an algorithm can be designed to check forall of the possible combinations of deletions that will eliminate all ofthe pathways capable of producing the desired objective.

[0107] (XXX—yes, insert details of the computational algorithm)

[0108] Together the formulation of the stoichiometric matrix andreaction constraints for a metabolic network, followed by thecalculation of the set of extreme pathways and lastly the identificationof the entire collection of minimal deletion sets terminates the process10 at an end state 26.

[0109] Thus by mathematically describing the reactions deemed presentwithin a cellular metabolic network and adding or removing constraintson internal and exchange fluxes (inputs) in the network it is possibleto (1) simulate a genetic deletion event or complete enzymaticinhibition and (2) simulate the presence or absence of metabolicresources available to the network perhaps to mimic its in vivoenvironment. Based on the resulting mathematical description of thenetwork the unique set of extreme pathways, which define the completerange of metabolic phenotypes and production capabilities of thenetwork, can be calculated. From these pathways particular fluxes and/orsets of fluxes can be deleted from the network to assess the ability ofthe network to produce the constituents of the biomass or any particularmetabolic objective. If the removal of a set of internal fluxes from thenetwork eliminates the ability of the network to produce the desiredmetabolic objective such as precursors for growth, and the cell can notobtain these precursors from its environment, then the removal of thisset of fluxes constitutes a minimal deletion set that has potential asan antimicrobial drug target or combination of targets. All of theseminimal deletion sets can be calculated based on the process describedherein and the appropriate genetic targets identified from the networkcomposition.

[0110] In addition to using these deletion sets for the identificationof antimicrobial drug targets, these deletion sets can be used to designand engineer cells to have desired metabolic characteristics andcapabilities that can be assessed using a pathway analysis. This hasclear potential in the metabolic engineering of bacterial strains to becompetitively co-cultured. Cells can thus be engineered to requirespecific growth environments and substrate dependencies. Opposingstrains can even be designed to be co-dependent through the eliminationof genes as guided by the process described herein of identifyingminimal deletion sets.

[0111] It is also possible to calculate all deletion sets that willrender the network to operate sub-optimally below a certain thresholdfrom the extreme pathways, not just deletion sets that result incomplete functional losses. Following the same procedure disclosedherein with only a slight modification does this calculation. Instead oflooking for deletion sets that eliminate all of the pathways thatgenerate the particular objective, only deletion set that eliminatepathways that generate the particular objective above a specified yieldare calculated.

[0112] As a stoichiometric matrix or a connectivity matrix can begenerated to describe biological networks in other areas such as signaltransduction, the identification of minimal deletion sets is applicablefor identify drug targets beyond metabolic networks, extending even intothe identification of critical links or sets of points in non-biologicalnetworks.

EXAMPLE 1

[0113] Minimal Deletion Sets in a Hypothetical Reaction Network

[0114] Using the process disclosed in FIG. 1, the entire collection ofminimal deletion sets has been calculated for a hypothetical reactionnetwork to completely illustrate the approach. The hypothetical reactionscheme considered is shown in FIG. 3(A). This network is comprised of 6metabolites, 8 internal fluxes, and exchange fluxes. Note that internalfluxes v₃ and v₄ may represent two separate reactions or be the resultof a decomposition of a reversible reaction into both a forward and areverse reaction. We will only allow metabolites A and C to enter thesystem as available substrates, and allow metabolite GRO to exit. GRO isa metabolite used to represent the result of a systemic demand (i.e. agrowth flux, v_(z)) of one mole of C and D and 2 moles of E (analogousto biosynthetic precursors used to generate biomass or actual componentsof the biomass).

[0115] After determining the reactions in the network and the exchangefluxes, all of the governing mass balance equations are assembled. Theseequations are provided in FIG. 3(B) and can be used to generate thestoichiometric matrix for the system. In addition all of the directionalconstraints on the reactions, due to systems specific conditions on theinput/output of various substrates and products and the thermodynamicsof the internal reactions, are also provided in FIG. 3(B). The nextstage in determining the minimal deletion sets for a metabolic networkis the identification of the set of extreme pathways for the network.For the system provided in FIG. 3(A) the calculated extreme pathways areprovided in FIG. 3(C). Pathway 1 and 2 use both A and C to reach thegrowth objective while pathway 3 and 4 only use C, and 5, 6, 7, and 8use only metabolite A as their input. Pathway 9 and 10 correspond tointernal cycles in the network and are ignored from furtherconsideration, as they do not add functionality to the network. Eachpathway satisfies the mass balances and flux constraints placed on thenetwork in FIG. 3(B)

[0116] As previously mentioned when a reaction is eliminated from thenetwork or identically if the flux value is forced to zero we can seefrom Equation 3 that each of the pathways that utilize the reaction areforced to be weighted to zero, otherwise non-zero values would exist inv corresponding to the eliminated reaction. Therefore all pathways thatuse this flux can be eliminated for further analysis of the situation.As an example consider the elimination of flux v₁ from the network inFIG. 3(A). We can seen from Table 1 that this would remove pathways 1, 2and 5 through 8, leaving only pathway 3 and 4 as feasible pathways thatcan operate in the network. This is logical as the elimination of thisparticular flux would not allow metabolite A to be utilized, andsubsequently all the pathways that are eliminated represent those thatutilize metabolite A.

[0117] Now if we consider the case that only metabolite C is availableto the network by forcing b₁ to equal zero, we are left once again withonly pathway 3 and 4. The minimal deletion sets for growth on anyparticular substrate or combination of substrates corresponds to all ofthe sets of reactions that when deleted eliminate all of the pathwaysthat are available to produce biomass (in our case demonstrated by apositive flux level on flux b₃). Thus for the situation of growth on Cwe can see that the elimination of fluxes v₁ through v₄ does not affectthe remaining pathways (p₃ and p₄). Eliminating reaction v₅ and v₈individually eliminates both of the remaining pathways while eliminatingv₆ and v₇ together eliminate both of the pathways. Thus the minimaldeletion sets for growth on substrate C is simply {(v₅), (v₈), (v₆,v₇)}. We can perform similar calculations for growth on substrate Aalone by focusing on pathways 5 through 8 to reveal that the minimaldeletion sets under this condition are {(v₁), (v₂), (v₈), (v₃, v₅), (v₆,v₇)}. For growth in the presence of both substrates we would have toconsider all eight pathways. In this case the minimal deletion sets are{(v₈), (v₁, v₅), (v₃, v₅), (v₆, v₇)}. Thus we examine a wide range ofsubstrate availability conditions using this approach and determine theminimal deletion sets for each particular condition. This process ofdetermining the minimal deletion sets from a set of vectors can beeasily automated on a computer to handle much larger reaction networksand sets of pathways representative of realistic cellular networks.

[0118] Linear optimization techniques such as those used in flux balanceanalysis (Varma, 1994, Edwards, 1999) can also be used to determine theresults of all the single and double deletion combinations to confirmour results. To consider the case that both A and C are available alinear programming problem can be set up to optimize for b₃ whileconstraining the exchange fluxes for A and C not to exceed the input ofmore than one mole of A and one mole of C into the system. The maximumvalue for b₃ in this case would equal 0.5. From this we can compare themaximum value of b₃ for each single deletion and double deletion case tothe “wild type” case. FIG. 4 displays the results of this type ofanalysis. There is no single reaction that is critical, but thecombinations of (v₁, v₅), (v₃, v₅), and (v₆, v₇) are lethal. Theseresults are in exact agreement with minimal deletion sets calculatedfrom the extreme pathways.

[0119] In this example the demand considered on the network was that ofthe GRO flux, but we could also consider a simpler case in which thedemand is only one metabolite such as D or E representative of aparticular cellular demand. The process is identical with the exceptionof the inequality constraints that are placed on the exchange fluxesbefore the extreme pathways are calculated.

[0120] In accordance with the preceding explanation, variations andadaptations of the mathematically based method of analyzing biochemicalreaction network, also know as metabolic networks, in accordance withthe present invention will suggest themselves to a practitioner of themathematical arts.

[0121] In accordance with these and other possible variations andadaptations of the present invention, the scope of the invention shouldbe determined in accordance with the following claims, only, and notsolely in accordance with that embodiment within which the invention hasbeen taught.

What is claimed is:
 1. In a method of analyzing the production of one ormore selected metabolites of a biochemical reaction network producingmetabolites, the method having as inputs reactions of the biochemicalreaction network constructed from genomic and biochemical data, exchangefluxes on such of the produced metabolites as are of interest as inputsand outputs to the network, a stoichiometric matrix, developed from thereactions in consideration of the exchange fluxes, definingparticipation of each network metabolite in each reaction and exchangeflux of the network, and a system of linear equations and inequalitiesmathematically defining the network, the method serving to identifydeletion sets of reactions that, when removed from the network,eliminate the capability of the network to produce a selectedmetabolite, an improvement to the method comprising: where linearequations and inequalities of the network mathematically form a convexsolution space called a flux cone, calculating generating vectors of theflux cone, which generating vectors are called extreme pathways; andusing the generating vectors called extreme pathways, to determine setsof reactions that, when deleted, diminish capability of the network toproduce an output metabolite of interest; wherein the determinedreaction sets correspond to critical reactions of the network which,when stopped, affect the capability of the network to produce the outputmetabolite of interest.
 2. The method according to claim 1 that, afterthe determining of sets of reactions, further comprises: selecting fromthe determined sets of reactions those sets that totally eliminate thecapability of the network to produce the output metabolite of interest;wherein the selected sets are called deletion sets because deletion ofthe reactions represented by the pathways of these sets suffices tototally eliminate the production of the output metabolite of interest bythe network.
 3. The method according to claim 1 wherein the calculatingof the generating vectors of the flux cone is by mathematical process ofconvex analysis.
 4. The method according to claim 3 wherein themathematical process of convex analysis comprises: calculating any of(i) a conical basis, (ii) a convex basis, (iii) a linear basis, or (iv)a combination of any of conical and convex and linear bases.
 5. Themethod according to claim 1 wherein at least some of the constructedreactions will have an associated constraint upon the direction in whichthe reaction can proceed.
 6. The method according to claim 1 wherein theoutput of interest consists of one or more functional properties ofinterest in the analyzed biochemical production network; wherein thereaction sets show how these one or more functional properties ofinterest can be diminished or eliminated.
 7. The method according toclaim 6 wherein the output of interest consists of one functionalproperty of interest in the analyzed biochemical production network;wherein the reaction sets show how this functional property of interestcan be diminished or eliminated.
 8. The method according to claim 1wherein the biochemical reaction network analyzed represents a diseaseproducing, pathogenic, organism; and wherein the metabolite of interestis necessary for survival of the pathogenic organism; and wherein themethod further comprises: using the reaction set to target developmentof a drug that, by obstructing those reactions of the pathogenicorganism that produce the metabolite necessary for survival of theorganism, serves to eliminate the pathogenic organism.
 9. A drugdeveloped in accordance with the method of claim
 8. 10. The methodaccording to claim 1 wherein the biochemical reaction network analyzedrepresents a disease producing, pathogenic, organism; and wherein themetabolite of interest, produced by the pathogenic organism, isdeleterious, inducing disease; and wherein the method further comprises:using the reaction set for targeting the development of a drug that, byobstructing those reactions of the pathogenic organism produce themetabolite that induces disease, serves to eliminate the deleterious,disease-causing, function of the pathogenic organism.
 11. A drugdeveloped in accordance with the method of claim
 10. 12. The methodaccording to claim 1 wherein the reaction network analyzed is anorganism producing both desired bio-molecules of value and undesiredbio-molecules of no value; and wherein the metabolite of interestproduced by the organism is of the undesired valueless bio-molecules;and wherein the method further comprises: using the reaction set tometabolically re-engineer the organism to fail of those reactions thatproduce the metabolite of that is undesired and valueless, thereineliminating production of undesired valueless bio-molecules whilepermitting production of desired valued bio-molecules.
 13. Ametabolically re-engineered organism developed in accordance with themethod of claim
 12. 14. The method according to claim 1 wherein thereaction network analyzed is an organism producing both desiredbio-molecules of value by multiple metabolic routes; and wherein themetabolite of interest is produced by one of the routes of the organism;and wherein the method further comprises: using the reaction set tometabolically re-engineer the organism to fail of those reactions thatproduce the metabolite of interest via the one route, therein byeliminating production of metabolite via this route, nonetheless thatthe metabolite is of value, leaving intact production of the samemetabolite by alternative ones of the multiple metabolic routes.
 15. Ametabolically re-engineered organism developed in accordance with themethod of claim
 14. wherein the using of the generating vectors is todetermine sets of reactions that, when deleted eliminate capability ofthe network to produce an output metabolite of interest;
 17. A method ofusing the mathematical process of convex analysis where a convex hulldefined and spanned by unique generating, or edge, vectors is analyzableto derive a particular solution that is, mathematically, a particularpoint described by a flux vector lying within the interior of the convexhull, to the purpose of identifying critical points in cellularmetabolic networks engaging in biochemical reactions so that, byidentification of some critical point, the particular set of complexpathways of biochemical reactions leading to this point may be betterunderstood so that, by better understanding this set of complexpathways, it may further be better understood which reactions, andassociated pathways, can be disrupted so as to defeat that the cellularmetabolic network should attain this critical point, the methodcomprising: using the convex hull to represent the capabilities of ametabolic genotype where the unique generating, edge, vectors thatdefine and that span the hull represent systemically independent extremepathways of the metabolic, life, processes of the metabolic genotype,and where every point in the hull is some non-negative combination ofthe unique generating, edge, vectors corresponding to the fact thatevery metabolic, life, process of the metabolic genotype is somecombination of the extreme pathways of these metabolic processes;mathematically solving the convex hull by process of convex analysis toderive a particular solution that represents a metabolic phenotype wherethe particular solution is, mathematically, a particular point describedby a flux vector lying within the interior of the convex hull; andrepeating the mathematically solving until a complete set of particularsolutions, corresponding to a set of flux vectors each lying within theconvex hull, is derived which set of solutions corresponds to all thepathways by which a particular metabolic phenotype is realized; whereinderivation of all pathways by which the particular metabolic phenotypeis realized is tantamount to recognition of all the biochemicalreactions that, as part of any pathways, lead to the particularmetabolic phenotype; and wherein recognition of all biochemicalreactions variously leading to the particular metabolic phenotypepermits better understanding of what biochemical reactions of themetabolic genotype can be in particular disrupted so as to cause thatthe metabolic genotype is unable to realize the particular solution. 18.The method according to claim 17 employed on the genotype of apathogenic, disease-causing, organism wherein the method furthercomprises: developing drugs that, by obstructing those biochemicalreactions of the genotype of the pathogenic organism that lead to aparticular, disease-inducing, solution of the genotype, serve toeliminate the deleterious, disease-causing, phenotype of the pathogenicorganism.
 19. A drug developed in accordance with the method of claim18.
 20. The method according to claim 18 employed on the genotype of anorganism producing both desired bio-molecules of value and undesiredbio-molecules of no value, wherein the method further comprises:metabolically re-engineering the organism so as to obstruct thosebiochemical reactions of the genotype of the pathogenic organism thatlead to that particular solution where the phenotype produces theundesired valueless bio-molecules, eliminating production of theseundesired valueless bio-molecules while permitting continued productionof desired valued bio-molecules.
 21. A metabolically engineered organismdeveloped in accordance with the method of claim
 20. 22. A method ofanalyzing a metabolic network comprising: identifying all biochemicalreactions occurring in the metabolic network, including any directionsthereof; specifying all exchange fluxes, including any associateddirectional restraints, attendant upon metabolites of the identifiedbiochemical reactions; creating a stoichiometric matrix where eachcolumn in the matrix corresponds to a reaction, or flux, and where eachrow corresponds to a different metabolite involved in the metabolicnetwork; wherein the created stoichiometric matrix represents, in allits columns and rows, the collective biochemical reactions, being a formof chemical conversion, and the collective cellular transport processesof the metabolic network, which cellular transport processes are how themetabolites enter and leave the metabolic network; combining alldirectional constraints on the exchange fluxes with the createdstoichiometric matrix to define the metabolic network as a system oflinear equations and linear inequalities; and analyzing the system oflinear equations and linear inequalities that jointly define themetabolic network by mathematical process of convex analysis.
 23. Themethod according to claim 22 wherein the metabolic network defined as asystem of linear equations and linear inequalities obeys the threeequations S·v=0 where S refers to the stoichiometric matrix of thesystem and v is the flux vector; v_(i)≧0, ∀i where v_(i) corresponds tothe flux value of the i^(th) reaction; and α_(i)≦b_(i)≦β_(i)
 24. Themethod according to claim 23 wherein the analyzing comprises: solvingthe equations 1-3 in convex space as a convex polyhedral flux cone inn-dimensional space emanating from the origin of the space.
 25. Themethod according to claim 24 wherein every point on the convexpolyhedral flux cone is represented by${C = {{v:v} = {\sum\limits_{l - 1}^{k}{\omega_{i}p_{i}}}}},{\omega_{i} \geq {0{\forall_{i}.}}}$


26. The method according to claim 25 wherein the analyzing furthercomprises: calculating extreme pathways in the metabolic network asbeing represented by a conical hull of the flux cone.
 27. The methodaccording to claim 26 that, after the calculating that is part of theanalyzing, further comprises: determining from the calculated pathwayscritical biochemical reactions, or sets of biochemical reactions, thatare required for the metabolic network to attain a particular objectiveas is represented by a particular point on the flux cone.